Quantum character varieties and braided module categories

Abstract

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants ∫ SA of a surface S, determined by the choice of a braided tensor category A, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for A, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided A-modules are objects of the torus category ∫T2A. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of A= Rep qG with the category Dq(G/ G) -mod of equivariant quantum D-modules. When G= GLn, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra SHq,t.